Sen. Lindsey Graham, R-S.C., told reporters after a Senate briefing Tuesday that White House officials seemed "well on their way" to winning consensus for a plan that would unite Republicans on the contentious issue.

Trump is due to give an address Thursday at 2:30 p.m., according to Reuters.

While details of the plan have not yet been revealed, it broadly focuses on two key areas: legal immigration and border security.

Far fewer green cards would be given to people with relatives already in the U.S. The diversity visa lottery, which offers green cards to citizens of countries with historically low rates of immigration to the U.S., would be eliminated. The number of migrants entering the country would be left roughly unchanged.

It would harden the border by building more of Trump's coveted southern border wall and improve inspections of goods and people at ports of entry to fight drug smuggling.

The senators spoke as they were leaving their weekly policy lunch at the Capitol where they heard from Trump's son-in-law and senior adviser, Jared Kushner, and hardline immigration adviser Stephen Miller - who have been working on the plan with other White House officials for a couple of months. However, some individuals reportedly left the meeting feeling uneasy about its prospects. And then, number two, what are you going to do with the people there?

In a memorial service for slain law enforcement officers Wednesday, Trump showed no signs that he would shift his tone about immigrants to build more moderate support.

"I don't think it's created to get Democratic support as much as it is to unify the Republican Party around border security", Republican South Carolina Sen.

A group of Central American migrants is questioned about their children's health after surrendering to U.S. Border Patrol Agents south of the U.S. -Mexico border fence in El Paso, Texas, U.S., March 6, 2019.

Appearing on Wednesday's edition of The Lead, the conservative commentator lit fire to the Kushner plan for not sufficiently addressing the issue of what to do about the large number of migrants at the border seeking admittance into the United States.

The bill would end asylum claims at the US border for migrants from the Northern Triangle of Honduras, Guatemala and El Salvador. "I am willing to put other immigration ideas on the table to marry up with this but what I am not willing to do is ignore this problem any longer".

"If we do these four things, then the incentives created by our laws will cease to exist, this humanitarian disaster will begin to fix itself and I am willing to sit down with Democrats and find a way to address the underlying problem in Central America", he said.

President Trump on Wednesday warned again about the dangers of undocumented immigrants, signaling no plans to temper his rhetoric even as he prepares to unveil a broad proposal aimed at balancing public perception of his administration's hard line agenda. However, many remain skeptical that major immigration reform - something that has befuddled Congress for years - has a chance of passing anytime soon.

The president has scheduled an afternoon speech from the White House Rose Garden where he will make his pitch.